15.13: Correlations

Correlation means that there is a relationship between two or more variables (such as ice cream consumption and crime), but this relationship does not necessarily imply cause and effect. When two variables are correlated, it simply means that as one variable changes, so does the other. We can measure correlation by calculating a statistic known as a correlation coefficient. A correlation coefficient is a number from -1 to +1 that indicates the strength and direction of the relationship between variables. The correlation coefficient is usually represented by the letter r.

The number portion of the correlation coefficient indicates the strength of the relationship. The closer the number is to 1 (be it negative or positive), the more strongly related the variables are, and the more predictable changes in one variable will be as the other variable changes. The closer the number is to zero, the weaker the relationship, and the less predictable the relationships between the variables becomes. For instance, a correlation coefficient of 0.9 indicates a far stronger relationship than a correlation coefficient of 0.3. If the variables are not related to one another at all, the correlation coefficient is 0.

The sign—positive or negative—of the correlation coefficient indicates the direction of the relationship. A positive correlation means that the variables move in the same direction. Put another way, it means that as one variable increases so does the other, and conversely, when one variable decreases so does the other. A negative correlation means that the variables move in opposite directions. If two variables are negatively correlated, a decrease in one variable is associated with an increase in the other and vice versa.

Examples of positive correlations are the relationship between an individual’s height and weight or the relationship between a person’s age and number of wrinkles. One might expect a negative correlation to exist between someone’s tiredness during the day and the number of hours they slept the previous night: the amount of sleep decreases as the feelings of tiredness increase. In a real-world example of negative correlation, student researchers at the University of Minnesota found a weak negative correlation (r = -0.29) between the average number of days per week that students got fewer than 5 hours of sleep and their GPA (Lowry, Dean, & Manders, 2010). Keep in mind that a negative correlation is not the same as no correlation. For example, we would probably find no correlation between hours of sleep and shoe size.

Correlations have predictive value. Imagine that you are on the admissions committee of a major university. You are faced with a huge number of applications, but you are able to accommodate only a small percentage of the applicant pool. How might you decide who should be admitted? You might try to correlate your current students’ college GPA with their scores on standardized tests like the SAT or ACT. By observing which correlations were strongest for your current students, you could use this information to predict relative success of those students who have applied for admission into the university.

Correlation Does Not Indicate Causation

Correlational research is useful because it allows us to discover the strength and direction of relationships that exist between two variables. However, correlation is limited because establishing the existence of a relationship tells us little about cause and effect. While variables are sometimes correlated because one does cause the other, it could also be that a third variable is actually causing the systematic movement in our variables of interest. For example, wealth may be positively correlated with intelligence, but that is likely because wealthy people can afford higher education, which in turn increases intelligence.

This text is adapted from OpenStax, Psychology. OpenStax CNX.

Sometimes, researchers may choose to more passively interact with a phenomenon, rather than to intervene and manipulate the behaviors of interest.

For instance, perhaps a scientist wants to know if there’s an association between eating a plant-based diet and sleep. In this case, she quantitatively measures two variables by asking participants to report how many vegetables they consumed that day, and later, how many hours they slept.

This design is known as correlational research—examining whether relationships exist between two variables.

After collecting both series of measurements, the researcher can visualize the data for each unit, in this case each person, on a graph—a scatterplot—with the variables—daily vegetable consumption and amount of sleep—placed on either axis.

Statistically, correlations are determined by calculating the correlation coefficient, commonly denoted as r—a number between -1 and 1 that indicates the direction, the sign, of the association and its overall strength, how tight the points align.

Here, the correlation could be positive, which means that the two variables move in the same direction. That is, people who consumed very few veggies slept less, whereas others who ate more, slept more.

In addition, the correlation could be strong, with a value close to 1, which indicates that the data points cluster linearly, with very few exceptions across individuals.

When the association has more exceptions, the linear pattern can disappear as the data points are spread out: the absolute value becomes closer to zero, and the correlation is considered weak to non-existent.

Now, if the two variables move in the opposite direction of each other, the correlation would be considered negative. That is, the more vegetables people consumed, the less they slept and vice versa. The strength is relatively strong, given the scatter is rather linear.

Importantly, correlation does not mean causation! Researchers would need to follow-up on observations to determine just how changes in one variable cause changes in another.

Positive Correlation – A relationship where both variables increase or decrease together. Correlation – A measurable relationship between variables that does not imply causation. Negative Correlation – A relationship where one variable increases while the other decreases. Correlation Coefficient – A value from -1 to +1 showing the strength and direction of correlation. Third Variable – An external factor that may explain the relationship between two correlated variables.

Define correlation – Explain the statistical relationship between variables and how it is measured. (e.g., correlation coefficient) Contrast positive vs. negative correlation – Show how the sign of the correlation indicates the direction of the relationship. (e.g., correlation sign) Explore predictive value – Explain how correlation helps in forecasting outcomes and supporting decisions. (e.g., test scores) Explain mechanism or process – Describe how to interpret the strength and direction of a correlation coefficient. Apply in context – Use real-world examples to show that correlation does not imply causation, highlighting potential third-variable influences.

Questions that this Positive Correlation will help you answer: What does a correlation coefficient represent? What is the difference between a positive and a negative correlation? Why does correlation not imply causation?

Students – Understand and interpret statistical relationships in psychology and research. Educators – Provides visual resource to explain correlation concepts and critical thinking. Researchers – Reinforces proper analysis and limits of correlational data. Science Enthusiasts – Encourages deeper thinking about patterns and predictions in data.